I have been thinking about my own question above.
A possible answer is as follows:
When we reduce the equation dp(x) = a'(x)b'(x) modulo we are then dealing with elements in the ring . That is, when we are dealing with the equation we are working with elements in .
When we argue that one of the two factors in the equation , say must be 0, we need to be sure that a'(x) and b(x) are not zero divisors and that means we need the ring to be an integral domain.
Can someone please confirm for me that my reasoning with respect to the proof of Gauss' Lemma is correct.